Traveling waves for a microscopic model of traffic flow

We consider the follow-the-leader model for traffic flow. The position of each car z_i(t) satisfies an ordinary differential equation, whose speed depends only on the relative position zi₊₁(t) of the car ahead. Each car perceives a local density ?_i(t). We study a discrete traveling wave profile W(x) along which the trajectory (?_i(t), z_i(t)) traces such that W(z_i(t)) = ?_i(t) for all i and t > 0; see definition 2.2. We derive a delay differential equation satisfied by such profiles. Existence and uniqueness of solutions are proved, for the two-point boundary value problem where the car densities at x ? ±? are given. Furthermore, we show that such profiles are locally stable, attracting nearby monotone solutions of the follow-the-leader model.